Determine an equation of the plane containing the points: .$\displaystyle A(2,0,2),\;B(4,3,1),\;C(0,1,3)$

Yes, we need a cross-product . . . but are you just guessing?. . Do you have any idea why?

To write the equation of a plane, we need a point on the plane, $\displaystyle (x_1,y_1,z_1)$. . (we have 3 to choose from), and the normal vector of the plane, $\displaystyle \vec n \,=\,\langle a,b,c\rangle$.
Then substitute into: .$\displaystyle a(x-x_1) + b(y - y_1) + c(z-z_1) \:=\:0$